Which of the following is always even?
Even numbers are multiples of 2.
\(2n\) is always even because it is 2 multiplied by an integer.
Algebraic Proof
Algebraic proof is used to show that statements are always true for all values. This topic focuses on logical reasoning and generalising patterns.
Overview
Algebraic proof means showing that a statement is always true by using algebra, not just by checking one or two examples.
Example claim: The sum of two consecutive numbers is always odd.
In a proof, you replace unknown numbers with algebraic expressions such as \(n\), \(2n\), or \(2n+1\), then simplify to show the result has the required form.
What you should understand after this topic
- What a mathematical proof means
- How to represent odd and even numbers algebraically
- How to represent consecutive numbers
- How to prove statements about multiples
- Why examples alone are not enough for a proof
Key Definitions
Proof
A logical argument showing a statement is always true.
Even Number
A number that can be written as \(2n\).
Odd Number
A number that can be written as \(2n + 1\).
Consecutive Numbers
Numbers next to each other, such as \(n\) and \(n+1\).
Multiple
A number written as another number times an integer.
General Form
An algebraic expression representing every number of a certain type.
Key Rules
Use general algebra
Do not just pick one or two example numbers.
Start with the correct form
Use \(2n\) for even and \(2n+1\) for odd.
Simplify fully
Rearrange the expression until the pattern is clear.
Match the target form
Show the final expression is odd, even, or a multiple by its structure.
Quick Structure Check
Even number
\(2n\)
Odd number
\(2n + 1\)
Two consecutive numbers
\(n\) and \(n+1\)
Three consecutive numbers
\(n\), \(n+1\), \(n+2\)
How to Solve
What is algebraic proof?
Algebraic proof is a way of showing that something is always true for all valid numbers. It is stronger than giving examples, because examples only show that something works sometimes.
Algebraic proof often involves expanding brackets and simplifying expressions carefully.
Exam tip: A proof is only complete when you clearly state a conclusion.
Key idea
A proof must work for every number in the pattern, not just the examples you tried.
How to represent number types
To prove statements, numbers must be written in algebraic form.
Why this matters: Using the correct algebraic form is essential for the proof to work for all numbers.
Even number
\(2n\)
Odd number
\(2n + 1\)
Consecutive numbers
\(n\) and \(n+1\)
Consecutive even numbers
\(2n\) and \(2n+2\)
How to structure an algebraic proof
Exam tip: Always finish with a sentence explaining what you have proved.
- Write the number or numbers in algebraic form.
- Carry out the required operation.
- Simplify fully using simplifying expressions.
- Show the final expression matches the target pattern.
- Write a clear conclusion.
Example Questions
Edexcel
Exam-style questions inspired by Edexcel GCSE Mathematics, focusing on forming general expressions and proving number results algebraically.
Edexcel
Write an even number in algebraic form.
Edexcel
Write an odd number in algebraic form.
Edexcel
Prove that the sum of two consecutive integers is always odd.
Edexcel
Prove that the sum of two even numbers is always even.
Edexcel
Prove that the difference between two odd numbers is always even.
AQA
Exam-style questions based on the AQA GCSE Mathematics specification, focusing on clear proof structure and reasoning with odd, even, and consecutive numbers.
AQA
Prove that the product of two consecutive integers is always even.
AQA
Prove that the sum of three consecutive integers is always a multiple of 3.
AQA
A number is written as \(2n+1\). Explain why this number must be odd.
AQA
Show that the square of an odd number is always odd.
AQA
Sam says, “The sum of two odd numbers is sometimes odd.” Use algebra to prove that Sam is wrong.
OCR
Exam-style questions aligned with OCR GCSE Mathematics, emphasising generalisation, proof, and explaining why examples alone are not enough.
OCR
Prove that the sum of an even number and an odd number is always odd.
OCR
Prove that the square of an even number is always a multiple of 4.
OCR
Two consecutive odd numbers are added together. Prove that the result is a multiple of 4.
OCR
Explain why checking one example does not prove that a statement is always true.
OCR
Prove that the sum of four consecutive integers is always even.
Exam Checklist
Step 1
Write the number type in algebraic form.
Step 2
Carry out the operation exactly as the question asks.
Step 3
Simplify or factorise until the structure is clear.
Step 4
State why the final form proves the claim.
Most common exam mistakes
Wrong form
Using \(n+1\) for odd instead of \(2n+1\).
Not enough algebra
Using examples instead of a general algebraic expression.
Incomplete proof
Stopping at the algebra without explaining what it means.
Pattern confusion
Mixing up ordinary consecutive numbers with consecutive even numbers.
Common Mistakes
These are common mistakes students make when writing algebraic proofs in GCSE Maths.
Using examples instead of algebra
Incorrect
A student tests a few numbers and concludes the statement is always true.
Correct
Testing examples is not a proof. Algebraic proof requires showing that the statement is true for all values using general expressions.
Using the wrong form for odd or even numbers
Incorrect
A student writes an even number as \(2n + 1\).
Correct
An even number should be written as \(2n\), while an odd number is \(2n + 1\). Using the wrong form leads to incorrect conclusions.
Not simplifying the expression fully
Incorrect
The student stops at an unsimplified expression that does not clearly show the result.
Correct
You must simplify the expression completely until the required form is clearly shown, such as a multiple of 2 or 3.
Not explaining the final result
Incorrect
The student finishes with an expression like \(2n\) but gives no explanation.
Correct
You must state why the final expression proves the statement, for example: “Since \(2n\) is even, the result is always even.”
Confusing consecutive numbers
Incorrect
A student writes consecutive even numbers as \(n\) and \(n+1\).
Correct
Consecutive integers are \(n\) and \(n+1\), but consecutive even numbers are \(2n\) and \(2n+2\). The correct form must be used.
Try It Yourself
Practise forming logical arguments using algebraic reasoning.
Foundation
Foundation Practice
Understand simple algebraic reasoning and patterns.
Write an expression for an odd number.
Start with an even number and add 1.
An odd number can be written as \(2n + 1\).
Which expression represents two consecutive integers?
Consecutive numbers differ by 1.
Consecutive integers are written as \(n\) and \(n + 1\).
Write expressions for two consecutive even numbers.
Even numbers go up in steps of 2.
Two consecutive even numbers are \(2n\) and \(2n + 2\).
Which expression is always divisible by 3?
Think about multiples of 3.
\(3n\) is always divisible by 3 because it is 3 multiplied by an integer.
Write an expression for three consecutive integers.
Start at n and add 1 each time.
Three consecutive integers are \(n, n+1, n+2\).
A student says \(2n + 1\) is even. What is wrong?
Test with n = 1.
\(2n + 1\) always gives an odd number, not even.
Write an expression for an even number plus 1.
Start with \(2n\).
\(2n + 1\) represents an odd number.
Which is always a multiple of 4?
Multiples of 4 are written as 4 times something.
\(4n\) is always divisible by 4.
Write an expression for two consecutive odd numbers.
Odd numbers increase by 2.
Two consecutive odd numbers are \(2n + 1\) and \(2n + 3\).
Higher
Higher Practice
Construct and understand full algebraic proofs.
Which expression proves that the sum of two even numbers is even?
Factorise the expression.
Since \(2n + 2m = 2(n + m)\), it is still a multiple of 2, so it is even.
Prove that the sum of two consecutive integers is odd. Write the final simplified expression.
Use \(n\) and \(n+1\).
\(n + (n + 1) = 2n + 1\), which is odd.
Which shows that the product of two even numbers is even?
Multiply the expressions.
\((2n)(2m) = 4nm = 2(2nm)\), so it is even.
Show that the difference between two consecutive integers is always 1. Write the simplified result.
Use \(n\) and \(n+1\).
\((n+1) - n = 1\).
A student claims that \(n^2\) is always even. Why is this incorrect?
Try n = 3.
If n is odd, then \(n^2\) is also odd, so it is not always even.
Show that the sum of three consecutive integers is divisible by 3. Write the simplified expression.
Use \(n, n+1, n+2\).
\(n + (n+1) + (n+2) = 3n + 3 = 3(n+1)\), so it is divisible by 3.
Which proves that an odd number squared is odd?
Expand the bracket.
\((2n+1)^2 = 4n^2 + 4n + 1 = 2(2n^2 + 2n) + 1\), which is odd.
Write an expression to prove that the sum of two odd numbers is even.
Write two odd numbers first.
\((2n+1)+(2m+1) = 2n + 2m + 2 = 2(n+m+1)\), which is even.
Which step is essential in algebraic proof?
Proof must work for all numbers.
Using a general variable ensures the proof works for all cases.
Show that an even number plus an odd number is odd. Write the simplified expression.
Use \(2n\) and \(2m+1\).
\(2n + (2m+1) = 2n + 2m + 1 = 2(n+m) + 1\), which is odd.
Games
Practise this topic with interactive games.
Games coming soon.
Frequently Asked Questions
What is algebraic proof?
Using algebra to show a statement is always true.
What is the key idea?
Generalise using variables instead of specific numbers.
What should I avoid?
Using only one example instead of proving generally.